Pis'ma v ZhETF, vol. 101, iss. 6, pp. 442-447
© 2015 March 25
Recovering of superconductivity in S/F bilayers under spin-dependent nonequilibrium quasiparticle distribution
I. V. Bobkova+*1\ A. M. Bobkov+
+ Institute of Solid State Physics of the RAS, 142432 Chernogolovka, Russia
* Moscow Institute of Physics and Technology 141700 Dolgoprudny, Russia
Submitted 22 December 2014 Resubmitted 9 February 2015
We study theoretically the influence of spin accumulation on superconductivity in a superconduc-tor/ferromagnet bilayer. It is well-known that the superconductivity in Superconductor/Ferromagnet (S/F) bilayers is suppressed by the proximity to a ferromagnet. The spin accumulation by itself is also a depairing factor. But here we show that creation of the spin accumulation on top of effective exchange depairing, caused by the proximity to a ferromagnet, can lead to an opposite result. The superconductivity can be partially recovered by spin-dependent quasiparticle distribution. The systems with realistic parameters are considered and the possible experimental setup is proposed.
DOI: 10.7868/S0370274X15060107
It is well-known that the Zeeman interaction of electron spins with magnetic or exchange field is destructive to singlet superconductivity. The behavior of a magnetic superconductor with an exchange field h was studied long ago [1-4]. It was found that homogeneous superconducting state becomes energetically unfavorable above the paramagnetic (Pauli) limit h = A/a/2. An inhomo-geneous state with a spatially modulated Cooper pair wave function (LOFF-state) can appear only in a narrow region of exchange fields exceeding this value, as it was predicted in [1, 2].
Superconductor/ferromagnet (S/F) hybrid structures also can behave analogous to magnetic superconductors. In particular, it was shown [5] that a thin S/F bilayer is equivalent to a magnetic superconductor in an effective exchange field. Another way to create an exchange field in a thin superconducting film is to contact it to a ferromagnetic insulator [6-10], as it was observed experimentally [9] and justified theoretically [10].
However, recently it was demonstrated [11] that the simultaneous applying of the exchange field and creation of spin-dependent quasiparticle distribution in such S/F heterostructures can lead to qualitatively new phenomenon. For a thin superconducting film the destructive effect of the exchange field can be fully compensated by the creation of spin-dependent quasiparticle distribution in it. This effect takes place even if the exchange field exceeds the paramagnetic limit consider-
e-mail: bobkova@issp.ac.ru
ably, that is under the condition that superconductivity of the equilibrium film is fully suppressed.
In [11] the effect was illustrated on the basis of a voltage-biased half metal/superconductor/half metal (HM/S/HM) heterostructure. A thin film (with the thickness less than the superconducting coherence length) is sandwiched between two half-metallic layers with opposite directions of magnetization. Half-metallic behavior has been reported in CrC>2 [12, 13] and in certain manganites [14]. In-plane effective uniform exchange field heg in the film is supposed to be created by spin-active interfaces with half metals. The spin-dependent quasiparticle distribution in the film can be generated by applying a voltage bias between the two half metals. In this case for spin-up subband the main voltage drop occurs at one of the HM/S interfaces, while for spin-down subband - at the other. As a result, the distribution functions for spin-up and spin-down electrons in the superconducting film are to be close to the equilibrium form with different electrochemical potentials. The superconducting order parameter becomes exactly equal to its value for zero exchange field when this difference in electrochemical potentials (spin imbalance) reaches heg.
For the considered nonequilibrium case the paramagnetic state cannot be realized because the distribution function is created and supported by the external conditions in such a way that the populations of majority and minority subbands in the film remain equal.
Here we demonstrate that the destructive effect of the exchange field can be compensated by the cre-
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ation of spin-dependent quasiparticle distribution and the superconductivity can be recovered not only for the HM/S/HM heterostructure, proposed in [11]. The point is that the experimental realization of such a structure is difficult at the moment. First of all, the magnetizations of the half metals should be strictly antiparallel, what is hard to reach experimentally. Second, the effective exchange field, induced in the superconductor, should be of the order of the zero-temperature superconducting gap in order to observe the effect. It also seems to be a problem to get in a controllable way such values of the effective exchange field due to proximity of a half metal.
The effect discussed here is basically the manifestation of the same superconductivity recovering, but it is considered for a system based on a S/F bilayer. The S/F bilayer is a well investigated system as theoretically, so as experimentally. For our purposes it is important that in S/F bilayers there is a mesoscopic analogue of the LOFF-state. This phenomenon was predicted theoretically [15, 16] and observed experimentally [17-21]. In this state Cooper pair acquires the total momentum 2Q or —2Q inside the ferromagnet as a response to the energy difference between the two spin directions. Here Q oc h/vf, where h is an exchange energy and vp is the Fermi velocity. Combination of the two possibilities results in the spatial oscillations of the condensate wave function ^(x) in the ferromagnet along the direction normal to the SF interface [22]. This oscillatory dependence is known to cause ir- Josephson junction formation [15, 17] and the non-monotonic (and, in particular, reentrant) dependence of the critical temperature of S/F bilayers on the F layer thickness [23-28]. The effect of superconductivity recovering can be observed in S /F bilayers just in this regime.
In order to create the appropriate spin-dependent distribution function it is enough to contact the S /F bilayer to a strong ferromagnet via a tunnel junction and to pass the electric current through the system. Such a setup is easy to realize experimentally in contrast to the system based on two half metals with strictly opposite magnetizations.
Now we turn to the detailed description of the proposed system and to the microscopic calculation. The sketch of the system under consideration is represented in Fig. 1. The S/F bilayer is a main part of the setup. It is composed of a singlet s-wave superconductor S and a weak ferromagnetic alloy F with the thicknesses ds and dp, respectively. The x-axis is normal to the bilayer plane and the F/S interface is at x = 0. The bilayer is sandwiched between the normal metal N and a strong ferromagnet F' (Fe, Ni, Co) via tunnel junctions. The system is biased by the voltage V in order to create
0
F
GFS
-dF 0 ds x
Fig. 1. Sketch of the system under consideration
the spin-dependent nonequilibrium distribution in the bilayer.
In our calculations we assume that: (i) the system is in the dirty limit, so the quasiclassical Green's function obeys Usadel equations [29]; (ii) the thickness of the S layer d,g < Here = ^/Ds/Aq is the superconducting coherence length, Ds is the diffusion constant in the superconductor and An is the bulk value of the superconducting order parameter at zero temperature. This condition allows us to neglect the variations of the superconducting order parameter and the Green's functions across the S layer; (iii) we work in the vicinity of the critical temperature, so the Usadel equations can be linearized with respect to the anomalous Green's function.
The retarded anomalous Green's function fR(e,x) is a 2 x 2 matrix in spin space. We assume that the exchange field in the F layer is homogeneous h = (0,0, h). In this case there are only singlet and triplet with zero spin projection on the quantization axis pairs in the system. In the language of Pauli matrices it means that fR{e,x) = [/f (1 + a3)/2 + /f (1 - CT3)/2]ia2, where <72,3 are the corresponding Pauli matrices in spin space. While we only consider the singlet pairing channel, the superconducting order parameter A = Aiao.
The linearized Usadel equation for the retarded anomalous Green's function where a =t, 4-> takes the form:
Ddlf* + 2*[e + ah(x)]f* - 2iA(x) =0. (1)
Here a = ±1 for /1(4.); D stands for the diffusion constant, which is equal to -Ds(f) hi the superconductor (ferromagnet); h(x) = h in the ferromagnet and h(x) = 0 in the superconductor. Analogously, A(x) = 0
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in the ferromagnet and A(x) = A in the superconductor.
Eq. (1) should be supplied by the Kupriyanov-Lukichev boundary conditions [30] at the S/F interface
(x = 0):
<TSdxf£s = "i ./ = Gfs (/«s - f«F)\x=0 , (2)
where o"s(f) stands for a conductivity of the S(F) layer and Gps is the conductance of the S/F interface. The boundary conditions at the ends of the bilayer are dxf£= = Here we neglect small
conductances G\o of the F'/F and S/N interfaces because they enter the resulting anomalous Green's function only as very small additional deparing factors.
Solving Eq. (1) under the assumption, that the anomalous Green's function weakly varies across the S layer, we obtain the anomalous Green's functions in the bilayer. In the S layer it take the form:
fa,s = "g'
£_„-_!__iGFSDsXa tanhfAq-rfp] ^
2<rS(is(Afftanh[Aff(iF] + GFS/<7F)'
where Aj. = —2i(e + ah)/Dp.
Due to the fact that the bilayer is thin as compared to the superconducting coherence length, the anomalous Green's function in it takes the form of Eq. (3), characteristic for a homogeneous superconductor. The denominator E of Eq
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